The spirals described on shells, and called concho-spirals,
are such as would result from winding plane logarithmic spirals on cones.

Henry Moseley (1801-1872)

Polar Equations

Definition of Polar Coordinates

To define polar coordinates, we first fix an origin O and an initial
ray from O.

Then each point P can be located by assigning to it a polar coordinate
pair (r, ø), in which the first number, r, gives the directed distance
from O to P and the second number, ø, gives the directed angle from
the initial ray to the segment OP:

Interesting Graphs

This investigation is going to explore several interesting graphs and their
associated polar equations.

Note: In the following polar equations, ø=t.

Spirals

Spirals of Archimedes

Polar graphs of the form r = at + b where a is positive and
b is nonnegative are called Spirals of Archimedes. They have the
appearance of a coil of rope or hose with a constant distance between successive
coils. The constant distance is .

The polar graph for

r = t + 2
for

is an example of a Spiral of Archimedes.

Consider
another example of a Spiral of Archimedes:

r = at

where 0<a<1 and b=0.

First, let a=0.1. So we get the equation r = 0.1t and its graph:

The graph represents that of a spiral.

Let a become smaller and tend to zero.

For example, when a=0.01, we get r=0.01t and its associated graph is also
a spiral.

For the polar equation r = at where a tends to be small,
the graph represents that of a spiral. As a becomes smaller and tends
to zero, the graph continues to become a tighter, more compressed spiral.

If we let a=0.00001 and magnify the graph of r = 0.00001, the graph still
represents a spiral.

Our original assumption, the number of petals is 2n, does not
hold true when n is odd. We can make a new generalization, however,
that does hold true. There is a difference between the number of rose petals
when n is odd and even. When n is odd, the number of petals
is n. When n is even, the number of rose petals is 2n.

What if we change cos to sin? Does the polar graph still represent a rose
curve? What effect does this change have on the polar graph?

Try graphing several of these polar equations.

r = 2 cos (3 t) (blue)

r = 2 sin (3 t) (purple)

r = 3 cos (4 t) (purple)

r = 3 sin (4 t) (red)

r = 4 cos (5 t) (purple)

r = 4 sin (5 t) (green)

Obviously, sin or cos can be interchanged and the polar graph still represents
that of a rose. Changing cos to sin, however, does rotate the rose.

Based on our observations, what generalizations can we draw about rose
curves?

In general, rose curves have equations of the form

r = a cos (nt)

r = a sin (nt)

where a>0 and n is a positive integer.

The length of each petal is a.

The number of leaves is determined by n.
If n is even, there are 2n petals.
If n is odd, there are n petals.

For Fun-- A Rose Within A Rose

Here is the equation: r = 1 - 2 sin (3 t)

The next equations are examples of several other interesting
polar graphs.